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Uncertainty Models in Computational Geometry

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Uncertainty Models in Computational Geometry Uncertainty Models in Computational Geometry A thesis submitted to fulfil requirements for the degree of Doctor of Philosophy Patrick Eades Faculty of Engineering The University of Sydney June 30, 2020 2 Abstract In recent years easily and cheaply available internet-connected devices have enabled the collection of vast amounts of data, which has driven a continued interest in efficient, elegant combinatorial algorithms with mathematical guaran- tees. Much of this data contains an inherent element of uncertainty; whether because of imperfect measurements, because the data contains predictions about the future, or because the data is derived from machine learning algorithms which are inherently probabilistic. There is therefore a need for algorithms which include uncertainty in their definition and give answers in terms of that uncertainty. Questions about the most likely solution, the solution with lowest expected cost or a solution which is correct with high probability are natural here. Computational geometry is the sub-field of theoretical computer science concerned with developing algorithms and data structures for geometric problems, that is problems involving points, distances, angles and shapes. In computational geometry uncertainty is included in the location of the input points, or in which potential points are included in the input. The study of uncertainty in computational geometry is relatively recent. Earlier research concerned imprecise points, which are known to appear somewhere in a geometric region. More recently the focus has been on points whose location, or presence, is given by a probability distribution. In this thesis we describe the most commonly used uncertainty models which are the subject of ongoing research in computational geometry. We present specific problems in those models and present new results, both positive and negative. In Chapter 3 we consider universal solutions, and show a new lower bound on the competitive ratio of the Universal Traveling Salesman Problem. In Chapter 4 we describe how to determine if two moving entities are ever mutually visible, and how data structures can be repeatedly queried to simulate uncertainty. In Chapter 5 we describe how to construct a graph on uncertain points with high probability of being a geometric spanner, an example of redundancy protecting against uncertainty. In Chapter 6 we introduce the online ply maintenance problem, an online problem where uncertainty can be reduced at a cost, and give an optimal algorithm. 3 4 Acknowledgements Acknowledgement and my gratitude are due first to my supervisor Juli´anMestre whose influence both as a mentor and as a coauthor cannot be overstated. This thesis would not have been possible without Juli´an,and his influence is apparent on every page of it. Likewise to my secondary supervisor Joachim Gudmundsson, who also served as my primary supervisor during Juli´an's sabbatical. Joachim's support has underpinned my entire time as a PhD student. Both personally and academically he has been a irreplaceable mentor. To my colleagues at the University of Sydney, especially to the Algorithms group, which has grown into the most enjoyable and engaging place one could imagine doing a PhD. Special thanks are due to Vikrant Ashvinkumar, Mica Brankovic, Ralph Holz, John Pfeifer, Andr´evan Renssen, Martin Seybold, William Umboh and Sampson Wong. To the members of the Geometric Computation Group of Utrecht University, who I was privileged to visit, and who strongly influenced me. Especially to my collaborators Ivor van der Hoog, Maarten L¨offlerand Frank Staals. To the anonymous examiners of this thesis for their time, their kind words and their helpful suggestions. To my parents for twenty eight years of unwavering love and support, to my friends for their love, and to Daniel, always. Patrick Sydney, June 30, 2020 5 6 Statement of Originality This is to certify that to the best of my knowledge, the content of this thesis is my own work. This thesis has not been submitted for any degree or other purposes. I certify that the intellectual content of this thesis is the product of my own work and that all the assistance received in preparing this thesis and sources have been acknowledged. Patrick Eades June 30, 2020 7 8 Authorship Attribution Chapters 3-6 of this thesis contain work developed and written in collaboration with my coauthors. In each case I was one of the main contributors of the paper. Authors are listed alphabetically, as is conventional in theoretical computer science. Chapter 3 will be published as: Patrick Eades and Juli´anMestre. An Optimal Lower Bound for Hierarchical Universal Solutions for TSP on the Plane. In The 26th International Computing and Combinatorics Conference (COCOON), Atlanta, USA, 2020. I was corresponding author and will present the work at the conference. Chapter 4 is published as: Patrick Eades, Ivor van der Hoog, Maarten L¨offler and Frank Staals. Trajectory Visibility. In The 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT), T´orshavn,Faroe Islands, 2020. I presented the work at the conference. Chapter 5 is being prepared for publication. It is joint work with Juli´anMestre. Chapter 6 is being prepared for publication. It is joint work with Vikrant Ashvinkumar, Maarten L¨offlerand Seeun William Umboh. In addition to the statements above, in cases where I am not the corresponding author of a published item, permission to include the published material has been granted by the corresponding author. Patrick Eades, June 30, 2020 As supervisor for the candidature upon which this thesis is based, I can confirm that the authorship attribution statements above are correct. Juli´anMestre, June 30, 2020 9 10 Contents 1 Introduction 15 1.1 Computational Geometry . 15 1.2 Uncertainty . 18 1.2.1 The Minimum Expected Weight Spanning Tree . 18 1.2.2 Uncertainty Models . 20 1.3 Summary of Contributions . 24 2 Related Work 29 2.1 The Traveling Salesman Problem . 30 2.2 Range Searching . 33 2.2.1 Orthogonal Range Searching . 34 2.2.2 Simplex Range Searching . 35 2.2.3 Semi-algebraic Range Searching . 36 2.2.4 Multi-level Data Structures . 36 2.3 Coresets and -nets . 37 2.4 Visibility . 38 2.5 Trajectories . 40 2.6 Trajectories and Visibility . 41 2.7 Spanners . 42 2.8 Uncertainty . 44 2.8.1 Clustering . 44 2.8.2 Minimum Spanning Tree . 45 2.8.3 Nearest Neighbours and Voronoi Diagrams . 45 2.8.4 Closest Pair . 46 2.8.5 Convex Hull . 46 2.8.6 Distance and Shape . 47 2.8.7 Fr´echet Distance . 47 2.8.8 Hyperplane Separability . 48 2.8.9 Range Queries . 48 2.8.10 -kernels . 48 2.8.11 Skylines . 48 2.8.12 Spanners . 49 2.8.13 Visibility . 49 11 12 CONTENTS 2.8.14 Dealing with Uncertainty . 50 3 The Universal Traveling Salesman 51 3.1 Background . 54 3.1.1 Previous Lower Bounds . 55 3.2 Preliminaries and Notation . 57 3.3 Logarithmic Lower Bound . 61 3.4 Conclusion . 68 3.4.1 Equal Measure . 68 3.4.2 Convex . 68 3.4.3 α-fatness . 68 4 Trajectory Visibility 69 4.1 Background . 70 4.2 Introduction . 73 4.2.1 Results . 74 4.3 Algorithms for testing visibility . 76 4.3.1 An O(n log n) time algorithm . 76 4.3.2 An Ω(n log n) lower bound. 78 4.3.3 A linear-time algorithm . 79 4.4 Semi-algebraic range searching . 82 4.4.1 Intersecting line segments with a quadratic curve segment 84 4.5 Intersecting a convex polygon with an algebraic curve . 91 4.6 A data structure for two entities moving inside a simple polygon 94 4.6.1 Querying the data structure . 95 4.7 Two moving entities crossing edges in a polygonal domain . 99 4.8 A data structure for queries with one moving entity . 102 4.8.1 Entity r is contained in a simple polygon . 102 4.8.2 Entity r can cross a simple polygon . 104 4.8.3 Polygonal domains . 108 4.9 Conclusions . 109 5 Uncertain Spanners 111 5.1 Background . 112 5.2 Problem Definition . 114 5.3 Algorithm . 115 5.3.1 Uncertain Spanners in Higher Dimensions . 117 5.3.2 Other Cone-Based Spanners . 118 5.4 A Lower Bound . 119 5.4.1 Order Statistics and Spacings . 120 5.4.2 Fractional Coupon Collecting . 121 5.4.3 Simulating the Discrete Problem with the Continuous . 122 5.4.4 Extensions . 123 5.5 Conclusion . 125 6 Maintaining the Ply of Unknown Trajectories 127 CONTENTS 13 6.1 Introduction . 127 6.1.1 Minimizing Ply . 127 6.1.2 Problem Statement . 128 6.1.3 Prior Work . 129 6.2 Optimal Play is NP-hard . 131 6.3 An Optimal Algorithm . 133 6.4 Lower Bound . 135 6.4.1 Lower Bound for ∆ = 1 . 135 6.4.2 General Lower Bound . 136 6.5 Conclusions . 139 7 Conclusion 141 14 CONTENTS Chapter 1 Introduction 1.1 Computational Geometry A defining trend of the last half-century has been incredible growth in the quantity of data collected, stored and analysed. This in turn has driven continued interest in efficient, elegant combinatorial algorithms with good asymptotic behavior and worst-case guarantees. Computational geometry is the field of computer science concerned with designing and evaluating algorithms for geometric problems. The geometry of a problem may come from the input objects (e.g. points, lines, regions) or from relationships between them (e.g. distance, angle, intersection.) In the same way that a graph is the fundamental object of combinatorics, a set of points embedded in Euclidean space is the fundamental object of computational geometry. Computational geometry is a relatively recent field within computer science.
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